A Cobordism–Branch Valence Framework for ABC

This Version 2 manuscript develops a cobordism–branch valence framework for studying height inequalities related to the ABC conjecture. Classical approaches typically rely on modularity, Arakelov geometry, Diophantine height theory, or transcendence methods. In contrast, this work introduces a geometric mechanism based on cobordism flows and branch–valence densities, informed by ideas from potential theory and optimal transport. The framework produces three–point ABC triangle inequalities with an explicit cobordism slack term, suggesting structural reasons such inequalities should hold in broad settings. This revision includes an expanded introduction, strengthened connections to established literature, and refined formulations of the geometric and transport perspectives. The author acknowledges the extensive assistance of the OpenAI language model ChatGPT (GPT-5.1), whose contributions included conceptual synthesis, organizational structuring, exposition refinement, and LaTeX drafting. All mathematical responsibility for correctness remains with the human author.